Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling (1005.2012v3)

Abstract: The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.

• The paper demonstrates that the dual averaging algorithm achieves reliable convergence by separating optimization steps from network deviations.
• It presents detailed theoretical analysis of convergence rates across various network topologies, with scaling laws derived using the spectral gap.
• Simulations and extensions to stochastic communication protocols confirm the method’s robustness and practical applicability in decentralized settings.

Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling

The paper "Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling" presents a comprehensive framework for decentralized optimization over networks. It examines the problem of efficiently optimizing a global objective, which is the sum of local convex functions, using only local computations and communications. The dual averaging methodology adapted for the distributed setting is rigorously analyzed, with convergence rates obtained as a function of network size and topology. This work situates itself within the context of significant practical applications, such as multi-agent coordination, sensor network estimation, and large-scale machine learning problems.

Introduction and Problem Formulation

The paper begins by outlining the problem of distributed convex optimization defined over a network, formalizing it as minimizing a global convex objective $f(x)$ subject to $x \in X$, where $f(x) = \frac{1}{n} \sum_{i=1}^n f_i(x)$. Each $f_i$ is a local convex function associated with node $i$ in a graph $G = (V, E)$. The nodes, corresponding to agents, can communicate only with their neighbors, thus imposing constraints on the algorithm's design.

The Dual Averaging Algorithm

The core of the proposed method lies in its dual averaging algorithm, which maintains a sequence of local parameters for each node, updated using local subgradient information. This technique leverages the concept of dual averaging, where subgradients are aggregated to form running averages, which are then used to update the local parameters. This approach separates the optimization steps from the effects of the network's communication constraints, allowing for a clear analysis of convergence influenced by the network's spectral properties.

Convergence Analysis

The convergence of the algorithm is established through a decomposition into optimization and network deviation terms. It is shown that the error in the solution can be bounded by terms related to the optimization algorithm and the network communication protocol. Specifically, the convergence rate is impacted by the spectral gap of the communication matrix $P$, which is indicative of the network's connectivity.

Theoretical Insights on Network Scaling

The analysis reveals that the number of iterations required by the algorithm is inversely proportional to the network's spectral gap. Consequently, the convergence rate is faster for well-connected networks. The theoretical predictions are confirmed by comparing convergence rates for various network topologies, such as cycles, grids, and expander graphs:

• Cycles and Paths: The convergence scales as $\mathcal{O}(n^2)$.
• Grids: The scaling is $\mathcal{O}(n)$.
• Expanders: These exhibit a desirable scaling of $\mathcal{O}(1)$, reflecting the efficient spread of information across the network.

Stochastic Communication and Randomized Protocols

The paper extends the framework to handle stochastic and time-varying communication matrices, making it applicable to more practical scenarios where not all nodes or edges are active at all times. This includes settings like gossip protocols and scenarios with potential node or link failures. For randomized communication, the authors derive convergence rates showing a dependence on the expected spectral properties of the random matrices, maintaining robust performance under such uncertainties.

Stochastic Gradient Scenarios

Furthermore, the algorithm is shown to be robust against noisy gradient information, which is critical in real-world applications where measurements or data are often imprecise. The inclusion of stochastic gradients does not significantly alter the main convergence results, demonstrating the algorithm's flexibility and robustness.

Simulation Results

Empirical simulations confirm the theoretical bounds derived in the paper. The performance of the algorithm on synthetic data and different network structures demonstrates that the theoretical analysis accurately predicts the behavior of the algorithm. The simulations validate the network scaling laws and illustrate the convergence properties under different topologies, firmly establishing the practical utility of the proposed method.

Implications and Future Directions

The results have significant implications for a broad range of applications in decentralized optimization and large-scale machine learning. By establishing a clear link between network properties and the convergence of distributed optimization algorithms, this paper provides a solid theoretical foundation for designing efficient algorithms for networked systems. Future research could explore further optimization methods adaptable to the dual averaging framework, offering more refined trade-offs between computational and communication efficiencies.

In summary, this paper delivers a meticulous treatment of dual averaging for distributed optimization, furnishing both theoretical guarantees and empirical validations. The insights into network scaling contribute to a deeper understanding of decentralized optimization in various practical domains, underscoring the importance of network structure in the efficacy of distributed algorithms.